Now the next quest - how to take them apart without damaging the shoulder and button...
These are not designed to be taken apart and, if you get new ones, you'll probably have to cut the stem to length so it's not a 5 minute replacement.
I suppose if your existing pusher is unserviceable and you think you might get somewhere with it, the following might help but, these parts are quite small and you might struggle so a watchmaker would be better. I believe you are very likely to end up with a completely broken pusher so, if the pusher works today, I would not try this yourself. Similar to the pictures above, here is a broken pusher and the cap is missing, as is the seal.
If I look at the ring (washer) in cross section, you'll see a leading chamfer.
Here is an intact pusher and you can see the ring is held in the pusher cap.
Here, I've added the other ring on top so you can see the relative diameters.
Initially, I would have thought that the caps were swaged or crimped onto the ring but looking at picture of
@michael e and my two above, it seems there is a groove in the cap and the cap and ring are pressed together so the cap is held by the interference of the ring in the cap. That would explain the chamfer on one side of the ring as it would be that side that is pressed in to the cap and the chamfer gives a "lead in". By the way, that would also explain why so many of these are missing the cap...
If you want to remove the cap, don't press on the pusher as it won't go far enough to remove the cap (see below). The only way I see is to put the pusher into a plate with a hole where the hole is just slightly larger than the outer diameter of the ring so the hole supports the circumference of the cap. Then grip the pusher part that goes in the case (the part of 2.5 mm diameter) with a pin vice and pull the pusher through the hole in the plate which should separate the ring from the cap. Then you'll end up with my first picture, but also a cap.
Now the problems start again. The pusher is a sealed unit and again has a ring to hold it all together - I am pretty sure this is swaged but it might be an interference fit again. You can see there is a wide washer below and you'll need to separate this from the pusher body to get to the internals - unless just replacing the gel seal that Michael shows resolves your issue - I don't see it will but, you can find out.
Re-fitting the ring to the cap is easy enough but, it will probably never be fully tight again unless you have a means to tighten the fit. At any point, it could separate and you'll lose the cap so, be aware of that.
Hope this helps, Chris